A calculus of the absurd

17.2.2 The properties which make modular arithmetic

The first core property is this

  • Theorem 17.2.2 If \(a \equiv _m b\) and \(c \equiv _m d\), then

    \begin{equation} a + c \equiv _m b + d \end{equation}

Proof: TODO

  • Theorem 17.2.3 If \(a \equiv _m b\) and \(c \equiv _m d\), then

    \begin{equation} ac \equiv _m cd \end{equation}

Proof: TODO

  • Example 17.2.1 Calculate

    \begin{equation} R_{10}(17^{17}) \end{equation}

This requires a bit of careful thinking about, but the result should be

\begin{align} 17^{17} &\equiv _{10} 7^{17} &\equiv _{10} 7 7^{16} \\ &\equiv _{10} 7 \left (7^2\right )^8 \\ &\equiv _{10} 7 \left (-1\right )^{8} \\ &\equiv _{10} 7 \end{align}